CounterexampleConstruction: AI-generatedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-27

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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Weak sparsity of a pair does not force directional sparsity

Statement refuted

Refuted claim: if a pair $(X, Y)$ is weakly $c$ -sparse, then $X$ is $c$ -sparse to $Y$ .

Facts & Assumptions

Given: $X = {x_{1}, x_{2}}$ , $Y = {y_{1}, y_{2}}$ , $c = \frac{1}{2}$ , and the edges $x_{1} y_{1}, x_{1} y_{2}$ only.

Counterexample

technique · direct

1.1

The ordered cross-edge count is $e (X, Y) = 2$ , so $e (X, Y) = 2 \leq \frac{1}{2} ∣ X ∣ ∣ Y ∣ = \frac{1}{2} \cdot 2 \cdot 2 = 2$ . Thus $(X, Y)$ is weakly $\frac{1}{2}$ -sparse.

givenalgebra

1.2

But $x_{1}$ has two neighbors in $Y$ , and $2 > \frac{1}{2} ∣ Y ∣ = \frac{1}{2} \cdot 2 = 1$ . So $X$ is not $\frac{1}{2}$ -sparse to $Y$ .

givenalgebra

2.1

Therefore weak sparsity does not imply directional sparsity.

step 1.1step 1.2∎

Depends on

Sparsity of one vertex set to another, and weak sparsity of a pair

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

6 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Shenwei Huang, Yiao Ju, and Yidong Zhou, Erdős-Hajnal beyond the five-vertex path, §2 (standard reference, not scraped)